Arch Calculator finds arc length, radius, segment area, angle, and balloon count from span and rise using circular arch formulas: R = W²/(8h)+h/2 and L = Rθ for layout or decor estimates.
The Arch Calculator finds circular arch dimensions from two physical measurements: arch span (the straight chord width between base points) and arch rise (the vertical height from the chord to the arch peak). From those inputs it calculates total arc length, arch radius, segment area, central angle, and an optional balloon count estimate for packed spiral designs. All results assume a circular arch segment. Use it to plan material lengths, verify arch proportions, or rough-estimate balloon quantities before ordering.
Understanding the Inputs
This arch span and rise calculator takes two required measurements and two optional settings. Span and rise must be entered in the same unit system you select.
The straight-line chord distance between the two base points of the arch, measured horizontally across the opening. This is the full opening width — not a curved length. For a doorway, it is the clear opening width. For a garden arch or tunnel, it is the ground-level distance from post to post.
The vertical distance from the chord line up to the highest point of the arch — also called the sagitta. For a symmetrical arch, measure straight up from the midpoint of the span to the peak of the curve. The ratio of rise to span determines whether the arch is a minor arc or a major arc.
Switches all displayed outputs between US Customary (inches and feet) and Metric (centimeters and meters). This changes units only — the underlying circular geometry and every formula result remain identical regardless of which system is selected.
Choose 5, 9, 11, or 12 inch balloons. This value is used only to estimate balloon count for packed spiral arch designs. It has no effect on arc length, radius, segment area, or central angle. The calculator always applies balloon diameter in inches for the estimate, regardless of the selected unit system.
Formulas and Geometry
Every output is derived from span (W) and rise (h) using standard circular segment geometry. The calculator detects whether the arch forms a minor arc (rise ≤ half the span) or a major arc (rise > half the span) and applies the appropriate central angle formula.
Arch Radius — from Span and Rise
$$R = \frac{W^2}{8h} + \frac{h}{2}$$ The sagitta formula. Returns the radius of the full circle of which the arch is one segment.Half Span
$$a = \frac{W}{2}$$ Half the chord width. Used as the argument for the arc-sine in the central angle calculation.Central Angle — Minor Arc (rise ≤ half span)
$$\theta = 2\sin^{-1}\left(\frac{a}{R}\right)$$ Produces an angle less than π radians (180°). Applies to most common shallow arch shapes.Central Angle — Major Arc Adjustment (rise > half span)
$$\theta = 2\pi - 2\sin^{-1}\left(\frac{a}{R}\right)$$ Produces an angle greater than π radians. Applied automatically when arch rise exceeds half the span.Arc Length
$$L = R\theta$$ The total curved length of the arch — the material length needed to span the curve from one base point to the other.Sector Area
$$A_{sector} = \frac{1}{2}R^2\theta$$Triangle Area (used in segment calculation)
$$A_{triangle} = \frac{1}{2}W|R-h|$$Segment Area — Minor Arc
$$A_{segment} = A_{sector} - A_{triangle}$$Segment Area — Major Arc
$$A_{segment} = A_{sector} + A_{triangle}$$ Segment area is the 2D region enclosed between the curved arc and the straight chord.Effective Balloon Cluster Spacing
$$S_{effective} = 0.8B$$ Each balloon in a packed spiral design occupies approximately 80% of its nominal diameter as effective cluster spacing.Estimated Quad Clusters
$$Q = \left\lceil \frac{L_{in}}{S_{effective}} \right\rceil$$ Ceiling function — rounds up to the next whole cluster. Arc length is always converted to inches (Lin) for this step, regardless of unit system.Estimated Balloon Count
$$N = 4Q$$ Four balloons per quad cluster. The result is always a multiple of four.Formula Variables
| Symbol | Meaning |
|---|---|
W |
Arch span — the straight chord width between the two base points |
h |
Arch rise — the sagitta height from the chord to the arch peak |
R |
Arch radius — radius of the full circle of which the arch is a segment |
a |
Half span — half the chord width; the arc-sine argument |
θ |
Central angle in radians — the angular sweep of the arch from base point to base point |
L |
Total arc length — full curved length of the arch in the selected unit |
B |
Selected balloon diameter in inches (5, 9, 11, or 12) |
Lin |
Arc length converted to inches — used only for the balloon count estimate |
Q |
Number of quad clusters — ceiling of arc length divided by effective spacing |
N |
Estimated total balloon count — four balloons per cluster |
Worked Example
The table below shows results using the tool’s default values: 72 in span, 24 in rise, US Customary units, and 11 inch balloons. Because the rise (24 in) is less than half the span (36 in), the calculator applies the minor arc formulas throughout.
| Input / Output | Value | Detail |
|---|---|---|
| Arch Span | 72.00 in | Chord width — also the chord length output |
| Arch Rise | 24.00 in | Sagitta — rise is less than half span (36 in), confirming minor arc |
| Balloon Size | 11 inch | Used for balloon count estimate only |
| Total Arc Length | 91.73 in | Curved material length of the arch |
| Arch Radius | 39.00 in | Full diameter: 78.00 in |
| Segment Area | 1,248.70 sq in | ≈ 8.67 sq ft — area between arc and chord |
| Balloon Count | 44 balloons | 11 quad clusters (4 balloons per cluster) |
| Central Angle | 134.76° | 2.35 rad — Minor Arc |
How to Read the Results
Each output card corresponds to a specific geometric property or practical estimate. The cards below match the layout in the tool.
The full curved distance along the top of the arch from one base point to the other. This is the measurement you use when purchasing or cutting arch material — pipe, cattle panel, conduit, PVC, rope, garland, string lighting, or balloon frame wire. Arc length is always longer than the span because it follows the curve rather than the straight chord.
Use this as the minimum material length before fabrication. Add allowance for end connections, ground anchors, or material lost at cuts.
Default: 91.73 inThe radius of the complete circle of which the arch is one segment. A smaller radius produces a tighter, more steeply curved arch; a larger radius produces a shallower one relative to the span.
The card also shows the full circle diameter (radius × 2). Use the radius to mark the arch curve on-site with a trammel, beam compass, or a string-and-pin layout.
Default: 39.00 in / diameter 78.00 inThe two-dimensional area of the circular segment — the region enclosed between the curved arc and the straight chord. Useful for estimating the face area of a curved panel, lattice, fabric, cladding, or planking that fills the arch segment.
This is not the open floor area beneath the arch. It is the segment area only: the region between the arc and the chord line.
Default: 1,248.70 sq in (≈ 8.67 sq ft)A rough planning estimate for a packed spiral balloon arch using quad clusters — groups of four balloons arranged around a frame. The estimate applies an 80% packing factor to the selected balloon diameter and divides the arc length in inches by that effective spacing.
This is a starting-point estimate only. Final quantities depend on balloon brand, actual inflation size, frame tube diameter, and how tightly you pack each cluster. Always order extra balloons to account for popping and fitting adjustments.
Default: 44 balloons (11 quads, 11 in)The angle at the center of the full circle swept from one arch base point to the other, shown in both degrees and radians. The card also displays the Curve Type label:
Minor Arc — angle below 180° (π rad). The arch is less than a semicircle, which is the case for most standard arch designs.
Major Arc — angle above 180° (π rad). The arch curves past the circle’s midpoint, meaning the rise exceeds half the span.
When This Calculator Works Best
Whether you need a quick arc length calculator for material planning or a full arch dimensions calculator to verify proportions, this tool returns reliable results for circular arch segment geometry in the following situations.
- Laying out a circular arch opening on-site by finding the exact radius from a known span and rise, then marking the curve with a trammel or string line
- Estimating the unbent length of a cattle panel, conduit, or PVC pipe before bending it into a garden arch, hoop-house, or tunnel structure
- Planning garland, ribbon, string lighting, or trim length needed to follow the curve of a decorative arch or window opening
- Checking arch proportions before fabrication — confirming radius, arc length, segment area, and central angle from a known span and rise
- Getting a rough balloon quantity estimate before sourcing supplies for a packed spiral balloon arch design
- Verifying curved casing or trim layout for arched windows, doorways, or cabinetry where the opening is defined by span and rise
Important Limitations
Review these points before using calculator results in a build or event project.
- All calculations assume a circular arch segment. Results do not apply to parabolic, elliptical, pointed (Gothic), catenary, or any other non-circular arch shape.
- This is a geometry calculator only — not a structural engineering tool. It does not calculate load capacity, material stress, deflection, bearing reactions, or any physical forces acting on the arch.
- For arches that support weight or form part of a building, fence, or publicly accessible structure, verify dimensions and material requirements against your project specifications or consult a qualified professional.
- The balloon count is a rough estimate for packed spiral quad designs using an 80% packing factor. Actual quantities vary with balloon brand, inflation pressure, frame tube diameter, and packing density. Treat the number as a planning starting point and order additional balloons as a buffer.
- The calculator does not account for end fittings, overlaps, ground stakes, or material wasted at cut ends. Add an appropriate allowance for your specific application and connection method.
References
The following sources support the formulas and unit conventions used in this calculator.
-
Circular Segment — Sagitta, Chord, Radius, and Segment Area Weisstein, E. W. “Circular Segment.” MathWorld — A Wolfram Web Resource. Available at mathworld.wolfram.com. Supports the sagitta formula R = W² / 8h + h / 2, the chord-radius relationship, and the sector-minus-triangle method for segment area used in this calculator.
-
Arc Length, Sector Area, and Radian Measure Standard analytic geometry and trigonometry texts, including Stewart, J. Precalculus: Mathematics for Calculus, 7th ed. (Cengage). Supports the arc length formula L = Rθ, sector area ½R²θ, the arc-sine central angle derivation, and the radian-to-degree conversion used in the central angle output.
-
Unit Conversions — Inches, Feet, Centimeters, Meters Thompson, A. & Taylor, B. N. (2008). Guide for the Use of the International System of Units (SI). NIST Special Publication 811. National Institute of Standards and Technology, U.S. Department of Commerce. Supports the inch-to-foot and centimeter-to-meter conversions applied when switching between US Customary and Metric output.
-
Balloon Count — Practical Estimating Convention The balloon count estimate is based on a spacing assumption common in balloon décor planning: each balloon in a packed spiral design occupies approximately 80% of its nominal inflated diameter as effective cluster width. This is a practical planning convention, not a formally published engineering standard. Results are estimates and should be treated as starting points for material ordering, not guaranteed quantities.